Knife Project (projected text) is not transforming with the mesh
It's just a vertex weight problem, I think, perhaps because 'Automatic Weights' doesn't cut it, here.



If the vertices illustrated above are assigned with weight 1 to the groups tri_R, main, and tri_L respectively, everything seems to work.

Maybe select vertices in  Edit Mode in a wireframe view, to ensure you get them all, and assign to groups with weight 1 in the Data > Vertex Groups panel, rather than painting, in this case.